Fast direct integral equation methods for the Laplace-Beltrami equation on the sphere

Integral equation methods for solving the Laplace-Beltrami equation on the unit sphere are presented and applied to the problem of point vortex motion. The Laplace-Beltrami equation is first posed on a simply connected domain on the sphere, then reformulated into an integral equation and discretized. The resulting linear system is solved by adapting current fast direct solvers from fully two and three dimensional problems to the surface of the sphere. The solution is achieved in O(N) operations, where N is the number of points lying on the contour of a single “island.” The performance of the solver is studied through several representative examples. To highlight the efficiency of the direct method for problems with multiple right hand sides, the solver is used to study point vortex motion. The relationship between the Laplace-Beltrami equation and the motion of a point vortex in the presence of coastlines is explained—both in terms of finding instantaneous streamlines of the fluid and the trajectory of a vortex over time. The solver is used to construct these instantaneous streamlines and trajectories, of which the latter requires the Laplace-Beltrami equation to be solved for each time step. In this case the performance of the direct solver is found to exceed previous iterative approaches using the fast multipole method. Lastly, the fast direct solver is adapted to the multiply connected case and several numerical examples are presented.